Existence of ground state solutions for a Choquard double phase problem

Abstract

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^μ}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where ${\mathcal{L}}_{p,q}^a$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<\mu <N$, $1<p<N$, $p<q<p+\frac{\alpha p}{N}$, $0\le a(\cdot )\in C^{0,\alpha }({\mathbb{R}}^N)$ with $\alpha \in (0,1]$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.